变分方法与交叉科学

变分原理是自然界事物遵守的客观法则,变分方法随着人类认识世界及改造世界的需要而产生发展起来.近代临界点理论在非线性科学研究中的突出作用充分展示了变分方法是研究自然科学、社会科学等交叉科学十分重要的工具和手段.文章举例简要介绍了变分方法在交叉科学研发中的重要作用.

Continuity of truncated Hardy-Littlewood maximal function

This paper focuses on the continuity of the truncated Hardy-Littlewood maximal function.We first show that the truncated Hardy-Littlewood maximal function is lower semi-continuous.Then by investigating the behavior of the truncated Hardy-Littlewood maximal function when the truncated parameterγchanges,we obtain an equivalent condition of the continuity of the truncated Hardy-Littlewood maximal function.

Mutants only Partially Represent Characteristics of Calcium-ReleaseActivated Calcium Channel Gating

Calcium-release-activated calcium(CARC)channels are one of the major pathways of calcium entry in non-excitable cells.Despite a decade or two of research,its regulatory mechanism is not yet thoroughly understood.The slow progress is due to the complexity of its pores(i.e.,Orai)on one hand and the difficulty in capturing its regulatory complex on the other hand.As a result,possible gating mechanisms have often been speculated by exploring the structure and properties of constitutive open mutants.However,there is much debate about how they can truly reflect the gating of CRAC channels under physiological conditions.In the present study,we combined molecular dynamics simulations with free energy calculations to study three dOrai mutants(G170P,H206A,and P288A),and further calculated their current-voltage curves.Results show that these constructs adopt different approaches to maintain their conductive state.Meanwhile they have unique pore structures and distinctive rectification properties and ion selectivity for cations compared to wild-type pores.We conclude that although the mutants may partially capture the gating motion characteristics of wild-type pores,the information obtained from these mutants is likely not a true reflection of CRAC channel gating under physiological conditions.

单参数离散时间非线性系统的可镇定性定理（英文）

本文对基本的离散时间非线性单参数随机系统建立了可镇定性定理.该定理推进了文献[1]的结果,进一步完善了关于离散时间自适应控制的反馈能力刻画.离散时间单参数系统可镇定的一个重要非线性临界常数是4,用以刻画关于幂函数类系统的反馈能力.而作为本文定理的应用,本文对一类典型的单参数离散时间非线性随机系统发现了新的可镇定临界常数2.

On Self-dual 4-dimensional Pseudo-Riemannian Manifolds

For a 4-dimensional Riemannian manifold(M,g),Atiyah et al.in[Proc.Roy.Soc.London Ser.A,1978,362(1711):425-461]used the kernel of twistor operator D to construct a distribution V(D)on the dual bundle of the anti-self-dual spinor bundle on M.Now V(D)forms an almost complex structure on dual bundle.Moreover,they showed that this almost complex structure is integrable if and only if M is self-dual.In this paper,we extend the construction of V(D)to 4-dimensional pseudo-Riemannian manifolds of signature(2,2).And we give a new method to prove the curvature condition in the integrability condition of V(D).Using this new method,we study the integrability conditions and structure of V(D)when the signature of g is(2,2).

Conditional Kernel Covariance and Correlation

The conditional kernel correlation is proposed to measure the relationship between two random variables under covariates for multivariate data.Relying on the framework of reproducing kernel Hilbert spaces,we give the definitions of the conditional kernel covariance and conditional kernel correlation.We also provide their respective sample estimators and give the asymptotic properties,which help us construct a conditional independence test.According to the numerical results,the proposed test is more effective compared to the existing one under the considered scenarios.A real data is further analyzed to illustrate the efficacy of the proposed method.

Area-minimizing Cones over Stiefel Manifolds

We study the area-minimization property of the cones over Stiefel manifolds V_(m)(F^(n))(F=R,C or H)and their products,where the Stiefel manifolds are embedded into the unit sphere of Euclidean space in a standard way.We will show that these cones are areaminimizing if the dimension is at least 7,using the Curvature Criterion of[Mem.Amer.Math.Soc.,1991,91(446):vi+111 pp.].This extends the results of corresponding references,where the cones over products of Grassmann manifolds were considered.

有关完全平方数的几个结论

作为一类特殊的正整数,完全平方数蕴含着许许多多优美的结论,无不凸显数论的精妙绝伦.本文,笔者通过对一个问题的思考、探究,给出了关于完全平方数的另外一些性质,供读者参考.